Finite Group Databasebeta

← C23×#12692 #19160318

Subgroups — C23×#12692 #19160318

Order 5888.

976 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#536C211yesmaximal, characteristic
2C2×C23×#352C214yesmaximal, complemented
3C23×#1785C211yesmaximal, characteristic
4C23×#1782C211yesmaximal, characteristic
5C22×#41×C23C2211yescharacteristic
6C23×Q8×C23C411yescharacteristic, complemented
7C23×#536—28no—
8C2×C23×C4⋊Q8—21no—
9C2×C23×#124—21no—
10C2×C23×#123C412yescomplemented
11C2×C23×#123C2212yes—
12C2×C23×C22⋊Q8—21no—
13C2×C23×C22⋊Q8—21no—
14C2×C23×#117C411yescharacteristic, complemented
15C2×C23×#101—21no—
16C2×C23×#100C2212yes—
17C2×C23×#99—21no—
18C2×C23×#99C2212yes—
19C23×#407—21no—
20C23×#381—21no—
21C23×#352—28no—
22C2×#41×C23—24no—
23C2×#41×C23—42no—
24C2×#41×C23—42no—
25C22×Q8×C23—21no—
26C22×Q8×C23—24no—
27C22×Q8×C23—21no—
28C22×Q8×C23—42no—
29C22×Q8×C23—42no—
30C22×Q8×C23D411yescharacteristic
31C22×D4×C23C2×C411yescharacteristic
32C22×D4×C23D411yescharacteristic
33C22×D4×C23C2311yescharacteristic
34C23×C4×C23C2×C411yescharacteristic
35C23×C4×C23C2×C412yes—
36C23×C4×C23D411yescharacteristic
37C23×C4×C23D411yescharacteristic
38C2×M16×C23—22no—
39C23×C4⋊Q8—28no—
40C23×#124—28no—
41C23×#123—24no—
42C23×#123—24no—
43C23×#123—24no—
44C23×#123D418yes—
45C23×C22⋊Q8—28no—
46C23×C22⋊Q8—28no—
47C23×#117—24no—
48C2×C4⋊C4×C23—22no—
49C2×C4⋊C4×C23—21no—
50C2×C4⋊C4×C23—22no—
51C2×C22⋊C4×C23—22no—
52C2×C22⋊C4×C23—22no—
53C2×C22⋊C4×C23—22no—
54C2×C22⋊C4×C23C2×C412yes—
55C2×C42×C23—21no—
56C23×#101—42no—
57C23×#100—24no—
58C23×#99—24no—
59C23×#99—42no—
60C23×C22⋊C8—42no—
61C23×#95—42no—
62C24×C23C22⋊C411yescharacteristic
63C24×C23C22×C411yescharacteristic
64C24×C23C2×D411yescharacteristic
65#41×C23—416no—
66C2×Q8×C23—24no—
67C2×Q8×C23—24no—
68C2×Q8×C23—22no—
69C2×Q8×C23—22no—
70C2×Q8×C23—41no—
71C2×Q8×C23—416no—
72C2×Q8×C23—42no—
73C2×Q8×C23—42no—
74C2×Q8×C23—42no—
75C2×Q8×C23—42no—
76C2×D4×C23—22no—
77C2×D4×C23—22no—
78C2×D4×C23—22no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C2×D4×C23—41no—
83C2×D4×C23—41no—
84C2×D4×C23C2×D414yes—
85C22×C4×C23—22no—
86C22×C4×C23—22no—
87C22×C4×C23—22no—
88C22×C4×C23—22no—
89C22×C4×C23—22no—
90C22×C4×C23—22no—
91C22×C4×C23—22no—
92C22×C4×C23—22no—
93C22×C4×C23—22no—
94C22×C4×C23—22no—
95C22×C4×C23—41no—
96C22×C4×C23—41no—
97C22×C4×C23—42no—
98C22×C4×C23—42no—
99C22×C4×C23—42no—
100C22×C4×C23C22⋊C411yescharacteristic
101C22×C4×C23C22⋊C411yescharacteristic
102C22×C4×C23C22⋊C414yes—
103C22×C4×C23C22⋊C411yescharacteristic
104C22×C4×C23C2×D411yescharacteristic
105M16×C23—42no—
106M16×C23—42no—
107C2×C8×C23—42no—
108C4⋊C4×C23—24no—
109C4⋊C4×C23—28no—
110C4⋊C4×C23—28no—
111C22⋊C4×C23—28no—
112C22⋊C4×C23—22no—
113C22⋊C4×C23—28no—
114C22⋊C4×C23—42no—
115C22⋊C4×C23—42no—
116C22⋊C4×C23C22⋊C414yes—
117C42×C23—24no—
118#12692C2311yesmaximal, characteristic, direct product factor, Sylow
119C23×C23—21no—
120C23×C23—21no—
121C23×C23—21no—
122C23×C23—21no—
123C23×C23—21no—
124C23×C23—21no—
125C23×C23—21no—
126C23×C23—42no—
127C23×C23—42no—
128C23×C23—42no—
129C23×C23C2×C22⋊C411yescharacteristic
130C23×C23C2×C22⋊C412yes—
131C23×C23#12012yes—
132Q8×C23—44no—
133Q8×C23—48no—
134Q8×C23—48no—
135D4×C23—44no—
136D4×C23—44no—
137D4×C23—44no—
138C2×C4×C23—28no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—24no—
146C2×C4×C23—24no—
147C2×C4×C23—21no—
148C2×C4×C23—21no—
149C2×C4×C23—21no—
150C2×C4×C23—44no—
151C2×C4×C23—44no—
152C2×C4×C23—44no—
153C2×C4×C23—44no—
154C2×C4×C23—42no—
155C2×C4×C23—42no—
156C2×C4×C23—44no—
157C2×C4×C23—82no—
158C2×C4×C23—84no—
159C2×C4×C23#9912yes—
160C2×C4×C23#9912yes—
161C2×C4×C23#12012yes—
162C8×C23—82no—
163C2×#536C2×C2311yescharacteristic
164C2×#352C2×C2314yescomplemented
165#1785C2×C2311yescharacteristic
166#1782C2×C2311yescharacteristic
167C22×C23—22no—
168C22×C23—22no—
169C22×C23—22no—
170C22×C23—42no—
171C22×C23—42no—
172C22×C23—42no—
173C22×C23—42no—
174C22×C23—42no—
175C22×C23—42no—
176C22×C23—42no—
177C22×C23—41no—
178C22×C23—81no—
179C22×C23—81no—
180C22×C23#37911yescharacteristic
181C22×C23C2×#9911yescharacteristic
182C22×C23C2×#9911yescharacteristic
183C4×C23—22no—
184C4×C23—22no—
185C4×C23—22no—
186C4×C23—44no—
187C4×C23—41no—
188C4×C23—44no—
189C4×C23—162no—
190C22×#41C22×C2311yescharacteristic
191C23×Q8C4×C2311yescharacteristic, complemented
192#536—28no—
193C2×C4⋊Q8—21no—
194C2×#124—21no—
195C2×#123C4×C2312yescomplemented
196C2×#123C22×C2312yes—
197C2×C22⋊Q8—21no—
198C2×C22⋊Q8—21no—
199C2×#117C4×C2311yescharacteristic, complemented
200C2×#101—21no—
201C2×#100C22×C2312yes—
202C2×#99—21no—
203C2×#99C22×C2312yes—
204#407—21no—
205#381—21no—
206#352—28no—
207C2×C23—21no—
208C2×C23—21no—
209C2×C23—42no—
210C2×C23—42no—
211C2×C23—81no—
212C2×C23#178111yescharacteristic
213C2×C23#212312yes—
214C2×#41—24no—
215C2×#41—42no—
216C2×#41—42no—
217C22×Q8—21no—
218C22×Q8—21no—
219C22×Q8—24no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8D4×C2311yescharacteristic
223C22×D4C2×C4×C2311yescharacteristic
224C22×D4D4×C2311yescharacteristic
225C22×D4C23×C2311yescharacteristic
226C23×C4C2×C4×C2311yescharacteristic
227C23×C4C2×C4×C2312yes—
228C23×C4D4×C2311yescharacteristic
229C23×C4D4×C2311yescharacteristic
230C2×M16—22no—
231C4⋊Q8—28no—
232#124—28no—
233#123—24no—
234#123—24no—
235#123—24no—
236#123D4×C2318yes—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239#117—24no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4C2×C4×C2312yes—
247C2×C42—21no—
248#101—42no—
249#100—24no—
250#99—24no—
251#99—42no—
252C22⋊C8—42no—
253#95—42no—
254C23#1269211yescharacteristic, direct product factor, Sylow
255C24C22⋊C4×C2311yescharacteristic
256C24C22×C4×C2311yescharacteristic
257C24C2×D4×C2311yescharacteristic
258#41—416no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—416no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4C2×D4×C2314yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4C22⋊C4×C2311yescharacteristic
294C22×C4C22⋊C4×C2311yescharacteristic
295C22×C4C22⋊C4×C2311yescharacteristic
296C22×C4C22⋊C4×C2314yes—
297C22×C4C2×D4×C2311yescharacteristic
298M16—42no—
299M16—42no—
300C2×C8—42no—
301C4⋊C4—24no—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C22⋊C4—28no—
305C22⋊C4—22no—
306C22⋊C4—28no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4C22⋊C4×C2314yes—
310C42—24no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23C2×C22⋊C4×C2311yescharacteristic
322C23C2×C22⋊C4×C2312yes—
323C23C23×#12012yes—
324Q8—44no—
325Q8—48no—
326Q8—48no—
327D4—44no—
328D4—44no—
329D4—44no—
330C2×C4—28no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—82no—
350C2×C4—84no—
351C2×C4C23×#9912yes—
352C2×C4C23×#9912yes—
353C2×C4C23×#12012yes—
354C8—82no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—41no—
365C22—42no—
366C22—81no—
367C22—81no—
368C22C23×#37911yescharacteristic
369C22C2×C23×#9911yescharacteristic
370C22C2×C23×#9911yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—44no—
375C4—41no—
376C4—44no—
377C4—162no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2—81no—
383C2C23×#178111yescharacteristic
384C2C23×#212312yes—
subgroupG/H
center, socleC22×C23C2×#99
commutatorC24C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC22×#41×C23C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C22×#41×C23 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C23×#352C2semidirect
6. C23×Q8×C23C4semidirect
10. C2×C23×#123C4semidirect
14. C2×C23×#117C4semidirect
118. #12692C23direct (N × H)
164. C2×#352C2×C23semidirect
191. C23×Q8C4×C23semidirect
195. C2×#123C4×C23semidirect
199. C2×#117C4×C23semidirect
254. C23#12692direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C23×#5362
C2×C23×#3522
C23×#17852
C23×#17822
#1269223